Transition Network Grammar

Transition Network Grammars developed out of the concept
of the transition network of a
finite-state automaton, but are
equivalent to push-down automata
because the arcs comprising
the network of a Transition Network Grammar
represent transcriptions of the rules of a context-free grammar
(Woods (1970)).
Sentences generated by the grammar are accepted
by a Transition Network Grammar through the process of
traversing the network comprised of these arcs.

The simplest form of Transition Network Grammar is called
a Recursive Transition Network (RTN).
RTNs form the core or basic component of the Augmented Transitions
Networks (ATNs).
An ATN is an RTN with additional resources that permit
it to process the complex attributes or features of lexical and syntactic
categories. Thus, for example, an ATN can perform subject/verb number
agreement.
ATNs are historically important, both computationally,
but also in terms of their application as psychological models of
language processing.
See, for example,
Fodor and Frazier (1980),
Wanner (1980), and
Wanner and Maratsos (1978).

This note deals only with Recursive Transition Networks, and the
following topics are discussed:

Recursive Transition Networks.
Recursive Transition Networks (RTNs)
resemble Finite-State and Push-Down Automata (FSAs and PDAs) in
several respects. In the first place, like FSAs and PDAs, RTNs are
recognisers (acceptors) of sentences generated by phrase-structure
grammars. In fact, RTNs are equivalent to PDAs in that they are capable
of accepting sentences generated by Chomsky Type 2 (context-free) grammars.
RTNs also resemble FSAs and PDAs in that their workings may be represented
employing transition networks. RTNs actually developed out of the concept
of the transition network associated with an FSA (and hence, their name).

Arcs and Nodes.
Beyond their pedigree, however, RTNs share little else with FSAs and differ
substantively from FSA transition networks. The most obvious of the
differences between the two classes of machine lies in the labelling of the
states (nodes) and arcs of their transition networks. While the states of
the transition network of an FSA are labelled by nonterminal symbols of the
grammar to which the automaton is equivalent, and the arcs are labelled by
the terminals, the arcs of an RTN are labelled by the nonterminals of the
equivalent grammar, and the nodes (states) are labelled arbitrarily, with
the labelling employed solely to distinguish among them. Thus, there is
normally no connection between the nodes of an RTN and the grammar to which
it corresponds (is equivalent).

Note that, although the foregoing observations cite FSA transition networks,
these statements apply equally to PDA networks, with the exception that
in the case of the latter machines, arcs corresponding to lambda/pop
transitions are not labelled by terminal symbols of the equivalent grammar.

Note also that, notwithstanding the statements above regarding the
labelling of RTN arcs only with nonterminal symbols, some special arcs can
be treated as if they were labelled with terminals. In some treatments of
RTNs, those arcs involved directly in the processing (reading and accepting)
of terminal symbols can be labelled with the terminal symbols. This issue
will be discussed below.

Subnetworks.
Another significant difference between RTNs and the transition networks of
FSAs and PDAs is that, while the networks of the latter machines consist of
one complete network, the network of the former can be treated as a
collection of subnetworks. There is one subnetwork for each distinct left-
hand side symbol in the rules of the production set of the grammar to which
the RTN is equivalent. For example, if the grammar includes the rewrite
rules

S ® NP VP,
NP ® DET N, and
VP ® V NP,

then the RTN will
consist of at least three subnetworks, one each for the symbols S, NP, and
VP. In some treatments of RTNs, there will also be subnetworks for the
preterminal, lexical category symbols DET, N, and V which appear on the
left-hand sides of the lexical rules of the grammar.

Paths.
The arcs in these subnetworks are labelled with the right-hand side symbols
of the rules to which the subnetwork corresponds. For example, there are
two arcs in the NP subnetwork corresponding to the two right-hand side
category labels of the rule

NP ® DET N

One of these is labelled DET while the other is labelled N.
These two arcs appear in the middle of the accompanying illustration of an
NP subnetwork.
The DET arc runs from the NP node to the node identified with the number 1.
The N arc extends from this node to the one labelled with the number 9
which identifies an accepting state for the subnetwork.
The two arcs
constitute a path through the NP subnetwork because the starting node of the
DET arc is the initial node of the subnetwork, and the ending node of the N
arc is the final, accepting node of the subnetwork.

Recall that the node labels are arbitrary.
Although the NP label is meaningful because it identifies the left-hand
of the rewriting rules to which the NP-subnetwork corresponds,
there is
no such rationale underlying choice of the labels for the other two nodes
except that they serve to distinguish among them.

Different paths in a subnetwork arise from the different rules
in the equivalent grammar with the same left-hand side symbol.
For example, if the production set of the grammar includes the three rules

VP ® V,
VP ® V NP, and
VP ® AUX VP,

with the left-hand side VP,
then the corresponding subnetwork will consist of three paths.

The topmost path in this illustration of a VP-subnetwork, consisting of the
one arc labelled V, corresponds to the rule VP ® V.
The middle path, consisting of the V and the NP arcs, corresponds to the
VP ® V NP rule,
while the bottom path, consisting of the AUX and VP arcs, corresponds to the
VP ® AUX VP rule.

Nondeterminism.
In connection with this VP-subnetwork, note that the RTN must be
nondeterministic because there are two arcs bearing the same label starting
from the initial VP node. Procedurally, nondeterministic behaviour of RTNs
is emulated by backtracking. If we assume arcs of the subnetwork are tested
in the order from top to bottom as indicated in the foregoing figure, then
the V arc terminating in the final state node of the subnetwork will be
tried first. Should this test fail, and result in the data string not being
accepted, then the RTN will, in effect, backup and try the second path, that
consisting of the V and NP arcs.

Recursion.
Note also in the subnetwork illustrated above that there is an arc, namely
the VP arc in the bottom path, the label of which is the same as the name of
the subnetwork itself. Thus, traversal of this arc requires re-entry into
the subnetwork containing it; that is, the recursive nature of the rule
VP ® AUX VP is translated into recursive traversal
of the corresponding subnetwork.
This recursive traversal of a subnetwork is actually the
origin of the use of "recursive" in the name Recursive Transition Network
for this class of recogniser.

Top-Down Processing.
An RTN performs a top-down recognition; that is, it begins by testing paths
of an S-subnetwork corresponding to those rules in the equivalent grammar
the left-hand side of which consists of the root or initial symbol. An
input string is accepted as a sentence generated by the equivalent grammar
if there exists a path in the S-subnetwork; that is, there exists a
sequence of contiguous arcs from the initial node to the final node of the
S-subnetwork.

A particular arc in a given subnetwork can be traversed only if there exists
a path in another subnetwork that has the same name as the label of the
the particular arc.
For example, the paths in the S subnetwork illustrated here correspond
to the rewrite rules

S ® NP VP,
S ® WH VP, and
S ® AUX NP VP .

The NP arc in the S subnetwork shown here, which corresponds to the
NP label on the right-hand side of the rule
S ® NP VP, can be traversed only if a path can
traversed the NP subnetwork.
Similarly, the VP arc in the S subnetwork can be traversed only if a path
in the VP subnetwork can be traversed.

Depth-First Processing.
The NP arc must, however, be traversed before the VP can be attempted.
Thus, the NP subnetwork must be entered, and one of its paths be
traversed, before an attempt to traverse the VP arc in the S subnetwork
can be undertaken.
Consequently, the NP subnetwork is entered, and the arcs comprising
one of its paths are traversed before a return is made to the S subnetwork.
The NP subnetwork path might consist of the DET and N arcs, meaning
that first a word in the determiner category, and then a word in the noun
category must be identified before any further arcs in the S subnetwork
can be traversed.
Processing by an RTN can therefore be described as proceeding
depth-first.

Cat Operation.
A cat operation is performed when an arc is traversed that is labelled
by a lexical category symbol.
For example, a "catting" operation is undertaken when the DET arc
in the NP subnetwork is traversed.
During this operation, a DET subnetwork is entered.
Subnetworks such as the DET, as well as any other subnetworks
corresponding to lexical rewriting rules, consist of arcs that
are labelled with words.
Thus, these subnetworks may be viewed as something like the
arcs in the transition network of a finite-state automaton.
If the word on the arc traversed is the same as a word in
the sentence being processed by the RTN, then that word
is accepted.

Push Operation.
The entry from one subnetwork into another is described as a push
operation.
For example, if an NP arc in the S subnetwork is being traversed,
and the RTN leaves the S subnetwork to enter the NP subnetwork,
then the RTN is described as "pushing for an NP."
The NP symbol in this case is pushed onto a push-down store to record
the arc of the S subnetwork that is being travered when the RTN leaves
it to enter the NP subnetwork.
On return to the S subnetwork from the NP subnetwork, if a noun phrase
has been identified, the NP label is popped from the store.

The push-down store is normally not included explicitly in the
specification of an RTN.
This store actually resides in the memory of the computer on which
the RTN is implemented.
It is this memory which enables a computer to keep track of its
position in a procedure, corresponding to the traversal of one
subnetwork, while it is performing another procedure that
corresponds to a different subnetwork.
This memory, in the form of a push-down store, provides RTNs
with the resources required to accept sentences generated by
context-free grammars.
Hence, recursive transition networks are equivalent to
context-free grammars, and to push-down automata.